import numpy as np
from matplotlib.animation import FuncAnimation
from matplotlib.animation import PillowWriter
from IPython.display import HTML, display
import matplotlib.pyplot as plt

def rect(x, a=1.0, b=2.0):
    # return ((x >= a) & (x <= b)).astype(float)
    return np.sin(b * x) + 1.0


def tri_decreasing(x):
    # g(x) = 1 - x on [0,1], else 0
    return np.where((x >= 0.0) & (x <= 1.0), 1.0 - x, 0.0)

t_min, t_max = 0.0, 3.0

tau = np.linspace(t_min, t_max, 2000)

f_tau = rect(tau)
y_conv = np.empty_like(tau)
for (i,t) in enumerate(tau):
    y_conv[i] = np.trapz(f_tau[:i] * tri_decreasing(t - tau[:i]), tau[:i])
    

tau[f_tau * tri_decreasing(1.5 - tau) > 0]

array([0.50125063, 0.50275138, 0.50425213, 0.50575288, 0.50725363,
       0.50875438, 0.51025513, 0.51175588, 0.51325663, 0.51475738,
       0.51625813, 0.51775888, 0.51925963, 0.52076038, 0.52226113,
       0.52376188, 0.52526263, 0.52676338, 0.52826413, 0.52976488,
       0.53126563, 0.53276638, 0.53426713, 0.53576788, 0.53726863,
       0.53876938, 0.54027014, 0.54177089, 0.54327164, 0.54477239,
       0.54627314, 0.54777389, 0.54927464, 0.55077539, 0.55227614,
       0.55377689, 0.55527764, 0.55677839, 0.55827914, 0.55977989,
       0.56128064, 0.56278139, 0.56428214, 0.56578289, 0.56728364,
       0.56878439, 0.57028514, 0.57178589, 0.57328664, 0.57478739,
       0.57628814, 0.57778889, 0.57928964, 0.5807904 , 0.58229115,
       0.5837919 , 0.58529265, 0.5867934 , 0.58829415, 0.5897949 ,
       0.59129565, 0.5927964 , 0.59429715, 0.5957979 , 0.59729865,
       0.5987994 , 0.60030015, 0.6018009 , 0.60330165, 0.6048024 ,
       0.60630315, 0.6078039 , 0.60930465, 0.6108054 , 0.61230615,
       0.6138069 , 0.61530765, 0.6168084 , 0.61830915, 0.6198099 ,
       0.62131066, 0.62281141, 0.62431216, 0.62581291, 0.62731366,
       0.62881441, 0.63031516, 0.63181591, 0.63331666, 0.63481741,
       0.63631816, 0.63781891, 0.63931966, 0.64082041, 0.64232116,
       0.64382191, 0.64532266, 0.64682341, 0.64832416, 0.64982491,
       0.65132566, 0.65282641, 0.65432716, 0.65582791, 0.65732866,
       0.65882941, 0.66033017, 0.66183092, 0.66333167, 0.66483242,
       0.66633317, 0.66783392, 0.66933467, 0.67083542, 0.67233617,
       0.67383692, 0.67533767, 0.67683842, 0.67833917, 0.67983992,
       0.68134067, 0.68284142, 0.68434217, 0.68584292, 0.68734367,
       0.68884442, 0.69034517, 0.69184592, 0.69334667, 0.69484742,
       0.69634817, 0.69784892, 0.69934967, 0.70085043, 0.70235118,
       0.70385193, 0.70535268, 0.70685343, 0.70835418, 0.70985493,
       0.71135568, 0.71285643, 0.71435718, 0.71585793, 0.71735868,
       0.71885943, 0.72036018, 0.72186093, 0.72336168, 0.72486243,
       0.72636318, 0.72786393, 0.72936468, 0.73086543, 0.73236618,
       0.73386693, 0.73536768, 0.73686843, 0.73836918, 0.73986993,
       0.74137069, 0.74287144, 0.74437219, 0.74587294, 0.74737369,
       0.74887444, 0.75037519, 0.75187594, 0.75337669, 0.75487744,
       0.75637819, 0.75787894, 0.75937969, 0.76088044, 0.76238119,
       0.76388194, 0.76538269, 0.76688344, 0.76838419, 0.76988494,
       0.77138569, 0.77288644, 0.77438719, 0.77588794, 0.77738869,
       0.77888944, 0.7803902 , 0.78189095, 0.7833917 , 0.78489245,
       0.7863932 , 0.78789395, 0.7893947 , 0.79089545, 0.7923962 ,
       0.79389695, 0.7953977 , 0.79689845, 0.7983992 , 0.79989995,
       0.8014007 , 0.80290145, 0.8044022 , 0.80590295, 0.8074037 ,
       0.80890445, 0.8104052 , 0.81190595, 0.8134067 , 0.81490745,
       0.8164082 , 0.81790895, 0.8194097 , 0.82091046, 0.82241121,
       0.82391196, 0.82541271, 0.82691346, 0.82841421, 0.82991496,
       0.83141571, 0.83291646, 0.83441721, 0.83591796, 0.83741871,
       0.83891946, 0.84042021, 0.84192096, 0.84342171, 0.84492246,
       0.84642321, 0.84792396, 0.84942471, 0.85092546, 0.85242621,
       0.85392696, 0.85542771, 0.85692846, 0.85842921, 0.85992996,
       0.86143072, 0.86293147, 0.86443222, 0.86593297, 0.86743372,
       0.86893447, 0.87043522, 0.87193597, 0.87343672, 0.87493747,
       0.87643822, 0.87793897, 0.87943972, 0.88094047, 0.88244122,
       0.88394197, 0.88544272, 0.88694347, 0.88844422, 0.88994497,
       0.89144572, 0.89294647, 0.89444722, 0.89594797, 0.89744872,
       0.89894947, 0.90045023, 0.90195098, 0.90345173, 0.90495248,
       0.90645323, 0.90795398, 0.90945473, 0.91095548, 0.91245623,
       0.91395698, 0.91545773, 0.91695848, 0.91845923, 0.91995998,
       0.92146073, 0.92296148, 0.92446223, 0.92596298, 0.92746373,
       0.92896448, 0.93046523, 0.93196598, 0.93346673, 0.93496748,
       0.93646823, 0.93796898, 0.93946973, 0.94097049, 0.94247124,
       0.94397199, 0.94547274, 0.94697349, 0.94847424, 0.94997499,
       0.95147574, 0.95297649, 0.95447724, 0.95597799, 0.95747874,
       0.95897949, 0.96048024, 0.96198099, 0.96348174, 0.96498249,
       0.96648324, 0.96798399, 0.96948474, 0.97098549, 0.97248624,
       0.97398699, 0.97548774, 0.97698849, 0.97848924, 0.97998999,
       0.98149075, 0.9829915 , 0.98449225, 0.985993  , 0.98749375,
       0.9889945 , 0.99049525, 0.991996  , 0.99349675, 0.9949975 ,
       0.99649825, 0.997999  , 0.99949975, 1.0010005 , 1.00250125,
       1.004002  , 1.00550275, 1.0070035 , 1.00850425, 1.010005  ,
       1.01150575, 1.0130065 , 1.01450725, 1.016008  , 1.01750875,
       1.0190095 , 1.02051026, 1.02201101, 1.02351176, 1.02501251,
       1.02651326, 1.02801401, 1.02951476, 1.03101551, 1.03251626,
       1.03401701, 1.03551776, 1.03701851, 1.03851926, 1.04002001,
       1.04152076, 1.04302151, 1.04452226, 1.04602301, 1.04752376,
       1.04902451, 1.05052526, 1.05202601, 1.05352676, 1.05502751,
       1.05652826, 1.05802901, 1.05952976, 1.06103052, 1.06253127,
       1.06403202, 1.06553277, 1.06703352, 1.06853427, 1.07003502,
       1.07153577, 1.07303652, 1.07453727, 1.07603802, 1.07753877,
       1.07903952, 1.08054027, 1.08204102, 1.08354177, 1.08504252,
       1.08654327, 1.08804402, 1.08954477, 1.09104552, 1.09254627,
       1.09404702, 1.09554777, 1.09704852, 1.09854927, 1.10005003,
       1.10155078, 1.10305153, 1.10455228, 1.10605303, 1.10755378,
       1.10905453, 1.11055528, 1.11205603, 1.11355678, 1.11505753,
       1.11655828, 1.11805903, 1.11955978, 1.12106053, 1.12256128,
       1.12406203, 1.12556278, 1.12706353, 1.12856428, 1.13006503,
       1.13156578, 1.13306653, 1.13456728, 1.13606803, 1.13756878,
       1.13906953, 1.14057029, 1.14207104, 1.14357179, 1.14507254,
       1.14657329, 1.14807404, 1.14957479, 1.15107554, 1.15257629,
       1.15407704, 1.15557779, 1.15707854, 1.15857929, 1.16008004,
       1.16158079, 1.16308154, 1.16458229, 1.16608304, 1.16758379,
       1.16908454, 1.17058529, 1.17208604, 1.17358679, 1.17508754,
       1.17658829, 1.17808904, 1.17958979, 1.18109055, 1.1825913 ,
       1.18409205, 1.1855928 , 1.18709355, 1.1885943 , 1.19009505,
       1.1915958 , 1.19309655, 1.1945973 , 1.19609805, 1.1975988 ,
       1.19909955, 1.2006003 , 1.20210105, 1.2036018 , 1.20510255,
       1.2066033 , 1.20810405, 1.2096048 , 1.21110555, 1.2126063 ,
       1.21410705, 1.2156078 , 1.21710855, 1.2186093 , 1.22011006,
       1.22161081, 1.22311156, 1.22461231, 1.22611306, 1.22761381,
       1.22911456, 1.23061531, 1.23211606, 1.23361681, 1.23511756,
       1.23661831, 1.23811906, 1.23961981, 1.24112056, 1.24262131,
       1.24412206, 1.24562281, 1.24712356, 1.24862431, 1.25012506,
       1.25162581, 1.25312656, 1.25462731, 1.25612806, 1.25762881,
       1.25912956, 1.26063032, 1.26213107, 1.26363182, 1.26513257,
       1.26663332, 1.26813407, 1.26963482, 1.27113557, 1.27263632,
       1.27413707, 1.27563782, 1.27713857, 1.27863932, 1.28014007,
       1.28164082, 1.28314157, 1.28464232, 1.28614307, 1.28764382,
       1.28914457, 1.29064532, 1.29214607, 1.29364682, 1.29514757,
       1.29664832, 1.29814907, 1.29964982, 1.30115058, 1.30265133,
       1.30415208, 1.30565283, 1.30715358, 1.30865433, 1.31015508,
       1.31165583, 1.31315658, 1.31465733, 1.31615808, 1.31765883,
       1.31915958, 1.32066033, 1.32216108, 1.32366183, 1.32516258,
       1.32666333, 1.32816408, 1.32966483, 1.33116558, 1.33266633,
       1.33416708, 1.33566783, 1.33716858, 1.33866933, 1.34017009,
       1.34167084, 1.34317159, 1.34467234, 1.34617309, 1.34767384,
       1.34917459, 1.35067534, 1.35217609, 1.35367684, 1.35517759,
       1.35667834, 1.35817909, 1.35967984, 1.36118059, 1.36268134,
       1.36418209, 1.36568284, 1.36718359, 1.36868434, 1.37018509,
       1.37168584, 1.37318659, 1.37468734, 1.37618809, 1.37768884,
       1.37918959, 1.38069035, 1.3821911 , 1.38369185, 1.3851926 ,
       1.38669335, 1.3881941 , 1.38969485, 1.3911956 , 1.39269635,
       1.3941971 , 1.39569785, 1.3971986 , 1.39869935, 1.4002001 ,
       1.40170085, 1.4032016 , 1.40470235, 1.4062031 , 1.40770385,
       1.4092046 , 1.41070535, 1.4122061 , 1.41370685, 1.4152076 ,
       1.41670835, 1.4182091 , 1.41970985, 1.42121061, 1.42271136,
       1.42421211, 1.42571286, 1.42721361, 1.42871436, 1.43021511,
       1.43171586, 1.43321661, 1.43471736, 1.43621811, 1.43771886,
       1.43921961, 1.44072036, 1.44222111, 1.44372186, 1.44522261,
       1.44672336, 1.44822411, 1.44972486, 1.45122561, 1.45272636,
       1.45422711, 1.45572786, 1.45722861, 1.45872936, 1.46023012,
       1.46173087, 1.46323162, 1.46473237, 1.46623312, 1.46773387,
       1.46923462, 1.47073537, 1.47223612, 1.47373687, 1.47523762,
       1.47673837, 1.47823912, 1.47973987, 1.48124062, 1.48274137,
       1.48424212, 1.48574287, 1.48724362, 1.48874437, 1.49024512,
       1.49174587, 1.49324662, 1.49474737, 1.49624812, 1.49774887,
       1.49924962])

# Figure setup
fig, axs = plt.subplots(2, 1, figsize=(8, 5))
axs[0].set_title("Input signals $f(\\tau)$ and $g(t-\\tau)$")
axs[1].set_title("Convolution output $\int_0^t f(\\tau) g(t-\\tau) d\\tau$")
t0 = 0.0
line_f, = axs[0].plot(tau, f_tau, lw=2, label='$f(\\tau)$')
line_g, = axs[0].plot(tau, tri_decreasing(t0 - tau), lw=2, label='$g(t-\\tau)$')
overlap = np.where(f_tau * tri_decreasing(t0 - tau) > 0)[0]
# fill_area = axs[0].fill_between(tau[overlap], 0, np.minimum(f_tau[overlap], tri_decreasing(t0 - tau)[overlap]), color='black', alpha=0.3)
line_y, = axs[1].plot(tau, y_conv, lw=2)
dot = axs[1].scatter(t0, 0.0, color='black')

axs[0].legend()
axs[0].set_ylim(0, max(f_tau)*1.2)
axs[1].set_ylim(0, max(y_conv)*1.2)

axs[1].set_xlabel("Time")
plt.tight_layout()


# Animation function
def update(frame):
    global dot
    # Shift and flip h
    h_shifted = tri_decreasing(frame - tau)
    # Convolution integral (numerical)
    # overlap = np.where(f_tau * h_shifted > 0)[0]
    dot.remove()
    # fill_area = axs[0].fill_between(tau[overlap], 0, np.minimum(f_tau[overlap], h_shifted[overlap]), color='black', alpha=0.3)
    conv_value = np.interp(frame, tau, y_conv)
    dot = axs[1].scatter(frame, conv_value, color='black')
    
    
    line_g.set_ydata(h_shifted)

    return line_y

ani = FuncAnimation(
    fig,
    update,
    frames=tau[::25],
    interval=50
)

HTML(ani.to_jshtml())

Once Loop Reflect

# Save the animation as a GIF using PillowWriter
writer = PillowWriter(fps=15, metadata=dict(artist='Me'), bitrate=1800)
ani.save('animated_sine.gif', writer=writer)

t = np.linspace(0, 20, 2000)
omega_n = 2.0
zeta = 0.3
K = 1.0
omega_d = omega_n * np.sqrt(1 - zeta**2)
step_response = 1 - np.exp(-zeta * omega_n * t) * (np.cos(omega_d * t) + (zeta * omega_n / omega_d) * np.sin(omega_d * t))
plt.figure(figsize=(8,5))
plt.grid(alpha=0.4)
plt.plot(t, step_response)
plt.savefig('step_response.svg', dpi=300)

t = np.linspace(0, 20, 2000)
omega_n = 2.0
zeta = 0.3
K = 1.0
omega_d = omega_n * np.sqrt(1 - zeta**2)
impulse_response = omega_n**2 / omega_d * np.exp(-zeta * omega_n * t) * np.sin(omega_d * t)
plt.figure(figsize=(8,5))
plt.grid(alpha=0.4)
plt.plot(t, impulse_response)
plt.savefig('impulse_response.svg', dpi=300, transparent=True)